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Quantitative Aptitude Mastery

The ultimate guide to cracking aptitude for Competitive Exams (SSC, Banking, Railway, TCS NQT, State Exams).

Quantitative Aptitude is the most critical section for competitive exams. This guide covers all essential formulas from Basic Arithmetic to Advanced Mathematics.

I. Number System & Basic Calculation

1. Number System (संख्या पद्धति)

  • Sum of first n natural numbers: n(n+1)/2
  • Sum of first n squares: n(n+1)(2n+1)/6
  • Sum of first n cubes: [n(n+1)/2]²
  • Divisibility Rules:
    • 3: Sum of digits divisible by 3.
    • 4: Last two digits divisible by 4.
    • 11: (Sum of odd place digits) - (Sum of even place digits) = 0 or multiple of 11.
  • Dividend Formula: Dividend = (Divisor × Quotient) + Remainder

2. HCF & LCM

  • Product Rule: 1st Number × 2nd Number = HCF × LCM
  • Fractions:
    • HCF of Fractions = (HCF of Numerators) / (LCM of Denominators)
    • LCM of Fractions = (LCM of Numerators) / (HCF of Denominators)

3. Simplification & Algebra

  • BODMAS: Bracket → Of → Division → Multiplication → Addition → Subtraction.
  • Important Identities:
    • (a+b)² = a² + b² + 2ab
    • (a-b)² = a² + b² - 2ab
    • a² - b² = (a+b)(a-b)
    • (a+b)³ = a³ + b³ + 3ab(a+b)
    • a³ + b³ = (a+b)(a² - ab + b²)
    • a³ - b³ = (a-b)(a² + ab + b²)

II. Arithmetic (Ankganit)

4. Percentage (प्रतिशत)

  • Percentage Change: [(Final - Initial) / Initial] × 100
  • Fraction Table (Must Memorize):
    1/2 = 50%1/3 = 33.33%1/4 = 25%
    1/5 = 20%1/6 = 16.66%1/8 = 12.5%
    1/10 = 10%1/20 = 5%

5. Profit, Loss & Discount

  • Profit %: (Profit / CP) × 100
  • Loss %: (Loss / CP) × 100
  • Selling Price (SP): CP × [(100 ± P/L%)/100]
  • Discount: Always calculated on Marked Price (MP).
  • SP (with Discount): MP × [(100 - D%)/100]
  • CP vs MP Relation: CP / MP = (100 - D%) / (100 + P%)

6. Simple & Compound Interest

  • Simple Interest (SI): (P × R × T) / 100
  • Compound Interest Amount (A): P(1 + R/100)ᵀ
  • Difference (CI - SI):
    • 2 Years: P(R/100)²
    • 3 Years: P(R/100)² × [(300 + R)/100]

7. Ratio, Proportion & Partnership

  • Mean Proportional (a, b): √(ab)
  • Third Proportional (a, b): b²/a
  • Partnership Rule: Profit Ratio = (Investment A × Time A) : (Investment B × Time B)

8. Average & Mixture

  • Average: Sum of observations / Number of observations
  • Average Speed: 2xy / (x+y) (For equal distances)
  • Alligation Rule:
    (Qty Cheaper) / (Qty Dearer) = (Price Dearer - Mean) / (Mean - Price Cheaper)

9. Time & Work

  • Work = Efficiency × Time
  • MDH Formula: (M₁×D₁×H₁) / W₁ = (M₂×D₂×H₂) / W₂
  • Shortcut: If A takes x days and B takes y days, Together = xy / (x+y) days.

10. Time, Speed & Distance

  • Distance = Speed × Time
  • Conversions:
    km/hr → m/s: Multiply by 5/18
    m/s → km/hr: Multiply by 18/5
  • Boats & Streams:
    • Downstream (D) = u + v
    • Upstream (U) = u - v
    • Boat Speed (u) = (D + U) / 2
    • Stream Speed (v) = (D - U) / 2

III. Advanced Maths

11. Advanced AIgebra

  • If x + 1/x = a, then x² + 1/x² = a² - 2
  • If x - 1/x = a, then x² + 1/x² = a² + 2
  • If x + 1/x = a, then x³ + 1/x³ = a³ - 3a
  • Quadratic Equation (ax² + bx + c = 0):
    Sum of roots (α+β) = -b/a
    Product of roots (αβ) = c/a

12. Mensuration 2D (Area)

  • Scalene Triangle: √[s(s-a)(s-b)(s-c)], where s=(a+b+c)/2
  • Equilateral Δ: Area = (√3/4)a²; Height = (√3/2)a
  • Circle: Area = πr²; Circumference = 2πr
  • Rhombus: Area = ½ × d₁ × d₂
  • Trapezium: Area = ½ × (Sum of parallel sides) × height

13. Mensuration 3D (Volume)

  • Cuboid: Vol = lbh; TSA = 2(lb+bh+hl)
  • Cube: Vol = a³; TSA = 6a²; Diagonal = a√3
  • Cylinder: Vol = πr²h; CSA = 2πrh
  • Cone: Vol = (1/3)πr²h; CSA = πrl; l = √(r²+h²)
  • Sphere: Vol = (4/3)πr³; TSA = 4πr²
  • Hemisphere: Vol = (2/3)πr³; TSA = 3πr²

14. Trigonometry

  • Basic: H² = P² + B² (Pythagoras)
  • sinθ=P/H, cosθ=B/H, tanθ=P/B
  • Values Table:
    θ0°30°45°60°90°
    sin01/21/√2√3/21
    cos1√3/21/√21/20
    tan01/√31√3∞
  • Identities:
    sin²θ + cos²θ = 1
    1 + tan²θ = sec²θ

15. Geometry

  • Sum of angles in a triangle = 180°
  • Circle Properties:
    • Angle in a semi-circle is 90°.
    • Tangents from an external point are equal.
    • Radius is perpendicular to Tangent.

16. Coordinate Geometry

  • Distance Formula: √[(x₂-x₁)² + (y₂-y₁)²]
  • Section Formula: x = (mx₂ + nx₁) / (m+n)
  • Slope (m): (y₂ - y₁) / (x₂ - x₁)

17. Modern Maths

  • Permutation (Arrangement): ⁿPᵣ = n! / (n-r)!
  • Combination (Selection): ⁿCᵣ = n! / [r!(n-r)!]
  • Probability: Favorable Outcomes / Total Outcomes
  • Coin Toss: Total outcomes = 2ⁿ

18. Data Interpretation (DI) Tips

"DI में कोई नया फॉर्मूला नहीं होता, इसमें Percentage, Ratio, और Average का तेज इस्तेमाल होता है।"

Rule of Thumb: 10% निकालना आसान है (बस एक दशमलव शिफ्ट करें), उसी के आधार पर बाकी गणना करें।